A Stochastic Lagrangian Proof of Global Existence of the Navier-Stokes Equations for Flows With Small Reynolds Number

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A stochastic Lagrangian proof of global existence of the Navier–Stokes equations for flows with small Reynolds number

Gautam Iyer

¹

Department of Mathematics, Stanford University, USA

Received 13 February 2007; received in revised form 23 September 2007; accepted 11 October 2007 Available online 1 November 2007

Abstract

We consider the incompressible Navier–Stokes equations with spatially periodic boundary conditions. If the Reynolds number is small enough we provide an elementary short proof of the existence of global in time Hölder continuous solutions. Our proof uses a stochastic representation formula to obtain a decay estimate for heat flows in Hölder spaces, and a stochastic Lagrangian formulation of the Navier–Stokes equations.

MSC:primary 76D03; secondary 76D05, 60K40

Keywords:Stochastic Lagrangian; Incompressible Navier–Stokes; Global existence

1. Introduction

The Navier–Stokes equations

∂_tu+(u· ∇)u−νu+ ∇p=0, (1.1)

∇ ·u=0 (1.2)

describe the evolution of the velocity field of an incompressible fluid with kinematic viscosityν >0. One of the (still open) million dollar problems posed by the Clay Institute [10] is to show that given a smooth initial datau0the solution to (1.1)–(1.2) in three dimensions remains smooth for all time.

In two dimensions, the long time existence of (1.1)–(1.2) is well known [3]. In three or higher dimensions, long time existence is known provided a smallness condition is imposed on the initial data. In 1962, Kato and Fujita [8]

showed existence of global solutions to (1.1)–(1.2) when the initial data is small inL³. Global existence when the initial data is small inH¹is standard and can be found in books. Possibly the most general result of this type (see Koch and Tataru [17]) shows global existence of (1.1)–(1.2) assuming that the initial data is small in BMO⁻¹. We also remark that recently Chemin and Gallagher [13] found a (non-linear) criterion on the initial data that guarantees global existence of (1.1)–(1.2), and does not reduce to a smallness criterion in BMO⁻¹.

E-mail address:[email protected].

1 The author was partly supported by NSF grant DMS-0707920.

doi:10.1016/j.anihpc.2007.10.003

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In this paper we prove global existence of (1.1)–(1.2) provided our initial data has small Hölder norm. Though global existence under our assumptions can be deduced from the Koch–Tataru result, the proof we present here (Sec- tion 5) is short, ‘elementary’ and essentially relies only on the decay of heat flows (Section 4), and a stochastic representation of the Navier–Stokes equations using particle trajectories (Section 3, see also [5,15]).

2. Notational conventions and description of results

In this section we describe the notational convention we use, and state the main result we prove. LetL >0 be a fixed length scale, andI= [0, L]. We define the Hölder norms and semi-norms onI^dby

|u|α= sup

x,y∈I^d

L^α|u(x)−u(y)|

|x−y|^α , uC^k=

|m|k

L^|^m^|sup

I^d |D^mu|, uk,α= uC^k+

|m|=k

L^k|D^mu|α,

whereD^mdenotes the derivative with respect to the multi indexm. We letC^kdenote the space of allk-times continu- ously differentiable spatially periodic functions onI, andC^k,αdenote the space of all spatially periodick+αHölder continuous functions. The spacesC^kandC^k,αare endowed with the norms · C^k and · k,α respectively.

We useIto denote the identity function onR^d(or onI^ddepending on the context), and useIto denote the identity matrix. The main theorem we prove in this paper is

Theorem 2.1.Letk1,α∈(0,1)andu₀∈C^k⁺^1,α(I^d)be spatially periodic, divergence free and have mean0. Let R=^L_νu₀k+1,α be the Reynolds number of the flow. Then∃T =T (k, α, d,_L¹u₀k+1,α)andR₀=R₀(k, α, d)such that for all0< R < R₀the solutionuof (1.1)–(1.2)with Reynolds numberR, initial datau₀and periodic boundary conditions is inC^k⁺^1,α for timeT, and satisfies

u_Tk+1,αu₀k+1,α. (2.1)

We prove Theorem 2.1 in Section 5. A few remarks are in order.

Remark 2.2.Local existence (Theorem A.5) combined with the Theorem 2.1 immediately show that for given initial data, we can chooseν large enough so that (1.1)–(1.2) have time globalC^k⁺^1,α solutions. Alternately for fixed viscosity, ifu₀k+1,αis small enough, Theorems 2.1 and A.5 again give time globalC^k⁺^1,α existence of (1.1)–(1.2).

Remark 2.3.The assumption thatu₀has mean 0 is not restrictive. First note that our boundary conditions imply that u_tis conserved in time. Setu¯=_L¹d

u₀to be the mean velocity. Now if we change to coordinates moving with the mean velocity by lettingu(x, t )=u(x+ ¯ut, t )− ¯u, thenusolves (1.1)–(1.2) with mean 0 initial datau₀− ¯u. Thus the smallness assumption in Remark 2.2 is really smallness assumptions on the deviation from the mean velocity.

Remark 2.4.Theorem 2.1 shows that for some timeT, Eq. (2.1) holds. Unfortunately our proof does not show that u_tk+1,αis decreasing in time.

3. The stochastic Lagrangian formulation

The Kolmogorov forward equation (or Feynman–Kac formula) [12,16] have been extensively used to represent solutions of linear parabolic PDE’s as the average of a stochastic process. In this section we briefly describe here a different approach used in [5,15,6] (see also [19,21]), which we use to provide a representation of the Navier–Stokes equations based on noisy particle paths.

Letu:R^d× [0,∞)→R^dbe some given (time dependent) vector field, andθa solution to the heat equation

∂_tθ+(u· ∇)θ−νθ=0 (3.1)

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with initial dataθ₀. We impose either periodic or decay at infinity boundary conditions onθ.

We expressθas the expected value of a stochastic process as follows: LetW be addimensional Wiener process, and letX:R^d→R^dbe a solution to the SDE

dX=u dt+√ 2ν dW

with initial data X₀(a)=a. Standard theory² [20] shows that the flow X is a homeomorphism, and as spatially differentiable asu. We letA_t denote the spatial inverse of the flow mapX_t.

Proposition 3.1.Ifu∈C¹,θ0∈C² then the unique solutionθ of (3.1)with initial dataθ0and either periodic or decay at infinity boundary conditions is given by

θ_t =Eθ0(A_t), (3.2)

whereEdenotes the expected value with respect to the Wiener measure.

Note that ifν=0, then Proposition 3.1 is nothing but the method of characteristics. Ifν >0, this can be interpreted as solving along random characteristics, and then averaging. Notice also that the Wiener process√

2νWtis the natural one to consider here, as it’s generator isν.

The reason we use the representation (3.2) and not the Kolmogorov forward equation is because the Kolmogorov forward equation in it’s natural setting involves final conditions, and not initial conditions. Thus the standard method employed by probabilists is to make at=T −s substitution [11]. The process obtained in this manner will have the same one dimensional distribution as the process A_t above, however spatial covariances and gradients of the two processes will in general be different. The stochastic representation of the Navier–Stokes equations we describe below involves spatial gradients of the flow mapA, and for this reason our representation will not be valid if we use the Feynman Kac formula.

We now use Proposition 3.1 to represent the solution to the Navier–Stokes equations as the expected value of a system that is nonlinear in the sense of McKean. The essential idea is to find a representation of the Euler equations involving particle trajectories [4], and then add noise and average as in Proposition 3.1 (as opposed to attempting to use the Kolmogorov forward equation).

Theorem 3.2.Letν >0,W be ann-dimensional Wiener process,k1and u0∈C^k⁺^1,α be a given deterministic divergence free vector field. Let the pairu,Xsatisfy the stochastic system

dX_t=u_tdt+√

2ν dWt, (3.3)

A_t=X⁻_t ¹, (3.4)

u_t=EP

(∇^∗A_t)(u₀◦A_t)

(3.5) with initial data

X(a,0)=a. (3.6)

We impose boundary conditions by requiringuandX−I are either spatially periodic, or decay at infinity. Thenu satisfies the incompressible Navier–Stokes equations(1.1)–(1.2)with initial datau₀.

Here Pin equation (3.5) denotes the Leray–Hodge projection onto divergence free vector fields [2] and ∇^∗A_t denotes the transpose of the Jacobian ofA_t. We remark that (3.5) is algebraically equivalent to

ω_t=E

(∇X_t)u₀

◦A_t, (3.7)

u_t= −⁻¹∇ ×ω (3.8)

and (3.5) can be replaced with (3.7)–(3.8) in Theorem 3.2. Note that (3.8) is exactly the Biot–Savart law. Whenν=0, Eq. (3.7) reduces to the well known vorticity transport for the Euler equations [2], and in this case (3.3)–(3.6) (or equivalently the system (3.3), (3.4), (3.6)–(3.8)) are exactly a Lagrangian formulation of the Euler equations [4].

2 See also [5,15,6] for an elementary proof for flows of the type we consider here.

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We do not prove Proposition 3.1 or Theorem 3.2 here, and we refer the reader to [5,15] instead. A generalization of Proposition 3.1 where the diffusion matrix is not spatially constant can be found in [15,6,19,21].

4. Decay of heat flows

In this section we prove a decay estimate for solutions to the heat equation with an incompressible drift. Our first estimate is anL^∞→L^∞estimate that is independent of the drift. A more generalL¹→L^∞version of this estimate appeared for example in [7] and [9]. We provide a proof that follows the proof in [7] and keeps track of the dependence of the constants on viscosity and our length scaleL.

Lemma 4.1.Letu∈C¹([0, T],I^d)be divergence free, andθbe a solution to Eq.(3.1)with initial dataθ0. Ifθ0is spatially periodic, mean0, and the dimensiond3, then there exists an constantc=c(d)such that

θ_t_∞ cL^d

(νt )^d/2θ0_∞.

Proof. Letϕbe mean zero and periodic,p^d⁺₄²andc=c(d, p)be a constant that changes from line to line. Then the Hölder, Poincaré and Sobolev in inequalities give

ϕ²=

ϕ^1/pϕ^(2p⁻^1)/pϕ^1/p_L1 ϕ^(2p_L(2p⁻−^1)/p1)/(p−1)cL^(4p⁻^d⁻^2)/2pϕ^1/p_L1 ∇ϕ^(2p_L2 ⁻^1)/p.

If we setq=_2p²₋₁this gives

∇ϕ²_L2cL^(qd⁻^4)/2ϕ²_L⁺2^qϕ⁻_L1^q.

Now letθandθto be solutions of (3.1) with initial conditionsθ₀⁻andθ₀⁺respectively. Integrating (3.1) immediately shows that

θand

θare conserved. Sinceθandθare of constant sign, this means thatθL¹ andθL¹

are conserved in time. Finally, the maximum principle implies thatθθθ, and henceθL¹ is nondecreasing in time.

Thus multiplying (3.1) byθand integrating overI^d gives

∂_tθ²_L2= −2ν∇θ²_L2−cνL^(qd⁻^4)/2θ⁻_L1^qθ²_L⁺2^q−cνL^(qd⁻^4)/2θ₀⁻_L1^qθ²_L⁺2^q.

Dividing byθ²_L⁺2^qand integrating in time gives θL²cL^2/q⁻^d/2

(νt )^1/q θ₀L¹.

LetPt(u)be the solution operator of (3.1). The above estimate shows Pt(u)

L¹→L²cL^2/q⁻^d/2 (νt )^1/q .

Sinceuis divergence free the dual operatorP^∗(u)=P(−u), and hence satisfies the same bound. Thus P2t_L¹_→_L∞Pt_L¹_→_L²Pt_L²_→_L∞= Pt_L¹_→_L²P_t^∗_(L∞)^∗→L²

PtL¹→L²Pt^∗L¹→L²cL^4/q⁻^d (νt )^2/q. Hence

θL^∞cL^4/q⁻^d

(νt )^2/qθ0L¹c L^4/q

(νt )^2/qθ0L^∞.

Finally,p^d⁺₄² is the same asq_d⁴, and choosingq=_d⁴ concludes the proof. 2

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Remark 4.2.Whend=2,p^d⁺₄²needs to be replaced withp > ^d⁺₄², and hence the above proof will show that for anyε >0,

θ_t_∞ cεL^d⁺^ε

(νt )^(d⁺^ε)/2θ₀_∞.

Lemma 4.1 cannot directly be used in our proof of Theorem 2.1. This is because the relation between the velocity fielduand the inverse flow map (Eq. (3.5)) involves the Leray–Hodge projectionP. The Leray–Hodge projection is a Calderón–Zygmund type singular integral operator and is known to be unbounded onC^k (see [22] for instance).

However, singular integral operators, and in particular the Leray–Hodge projection is known are known to be bounded on Hölder spaces, and thus we now proceed to extend Lemma 4.1 to Hölder spaces.

Using the stochastic flows from [5,15] we obtain the Hölder space estimate we need in an elementary manner.

We remark that the usual PDE methods [18] will only provide Hölder estimates that grow exponentially in time. The estimate we provide here will in general also grow exponentially in time, however decays in time when the viscosity is large, or driftUis small.

Lemma 4.3.Letd3andu∈C^k⁺^1,α([0, T],I^d)be divergence free and defineUby U= sup

t∈[0,T]utk+1,α. (4.1)

Letθ0∈C(I^d)have mean 0, andθsatisfy Eq.(3.1)with initial dataθ0. Then there existsT=T(^U_L, d, k, α)and a constantc=c(^{U T}_L , d, k, α)such that

θ_tk+1,αc L^d

(νt )^d/2+ U t

L

α

θ₀k+1,α

holds for allt∈ [0, T]. Ifd=2, the above estimate is still true if we replace_{(νt )}^L^d_d/2 with _{(νt )}^c^ε^L_(d+ε)/2^d+ε for anyε >0.

Remark 4.4.Note that the growth term is independent of the viscosity, and the decay term is independent of the driftu.

Proof. We present the proof ford3. Thed=2 case will then follow by replacingd withd+ε. DefineX, Aby Eqs. (3.3) and (3.4) respectively. From [5,15] and uniqueness of strong solutions to (3.1) we know

θt =Eθ0◦At.

Let =A−I be the Lagrangian displacement. First notice that iff∈C^0,αthen Lemma A.4 shows

|f ◦A_t|αc|f|α

U t L

α

a.s. (4.2)

Now, letma multi index with 1|m|k. We note thatD^mθ_t is a sum of terms of the form Dⁿθ0|A_t

1i|n|

Dⁿⁱ t and Dⁿθ0|A_t, (4.3)

whereni’s are multi indices with |ni|1 and |n| +

i|ni| = |m|. By Proposition 3.1 we know that EDⁿθ0|A_t

satisfies (3.1) with initial dataDⁿθ0, and hence by Lemma 4.1 we know E[Dⁿθ₀] ◦A_t

L^∞ cL^d

(νt )^d/2Dⁿθ₀L^∞. Thus using Lemma A.4, inequality (4.2) we have

E[Dⁿθ₀] ◦A_t

0,αc L^d

(νt )^d/2+ U t

L

α

Dⁿθ₀0,α. (4.4)

Using (4.4) and Lemma A.4, we bound the remaining terms of (4.3), concluding the proof. 2

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5. Global existence

In this section we prove Theorem 2.1. We start with a lemma involving bounds for the Leray–Hodge projection.

Lemma 5.1. Letk1, andA, v∈C^k⁺^1,α be such that∇A,v are spatially periodic. There exists a constantc= c(d, α)such that

P

(∇^∗A)v

k+1,αc∇Ak,αvk+1,α.

Proof. SincePvanishes on gradients, we can ‘integrate by parts’ to avoid the loss of derivatives. Note P

(∇^∗u)v

=P

∇(u·v)−(∇^∗v)u

= −P

(∇^∗v)u for anyu, v∈C¹. Thus we have

∂iP

(∇^∗A)v

=P

(∇^∗A)∂iv

−P

(∇^∗v)∂iA .

SincePis Calderón–Zygmund singular integral operator, it is bounded on Hölder spaces [22,1]. Finally note that the right hand side only depends on first derivatives ofAandv, and the lemma follows by taking Hölder norms. 2

We now prove Theorem 2.1. We restate it here for the readers convenience.

Theorem 2.1.Letk1,α∈(0,1)andu0∈C^k⁺^1,α(I^d)be spatially periodic, divergence free and have mean0. Let R=^L_νu0k+1,α be the Reynolds number of the flow. Then∃T =T (k, α, d,_L¹u0k+1,α)andR0=R0(k, α, d)such that for all0< R < R0the solutionuof (1.1)–(1.2)with Reynolds numberR, initial datau0and periodic boundary conditions is inC^k⁺^1,α for timeT, and satisfies

u_Tk+1,αu₀k+1,α. (2.1)

Proof. We assume thatd 3. Thed =2 case follows similarly by replacingd withd+ε. LetC,δbe the dimensional constants in Theorem A.5. We letU=Cu₀k+1,α, and chooseT such that ^{U T}_L < δ. By Theorem A.5, there exist a pair ofC^k⁺^1,α functionsX, u:[0, T] →I^d which are the unique (strong) solution to (3.3)–(3.6). Recall that u_tk+1,αUfor allt∈ [0, T].

From equation (3.5) we see

u_t=EPu0◦A_t+EP(∇^∗ )u0◦A_t.

Letc=c(k, α, d)be a constant that changes from line to line. Applying Lemmas 5.1 and A.4 to the second term we have

u_tk+1,αcEu0◦A_tk+1,α+cE∇^∗ tk,αu₀◦A_tk+1,α

and hence by Lemma 4.3 we have utk+1,αc

L^d (νt )^d/2+

U t L

α

u0k+1,α.

Minimizing _{(νt )}^L^d_d/2 +(^{U t}_L)^αin time shows that the minimum value is attained att₀=^cL_UR^d/(2α⁺^d), and the minimum value iscR^αd/(2α⁺^d). Thus we can chooseRsmall enough to ensuret0< T and equation (2.1) is satisfied. 2 Acknowledgement

I would like to thank Lenya Ryzhik for pointing out [7] which provides the elegant proof of Lemma 4.1 reproduced here.

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Appendix A. Bounds for the Lagrangian displacement

In this section, we prove bounds on∇X−Ik,α. The estimates proved here are elementary, and are taken directly from [14,15]. We reproduce them here for completeness and the readers convenience.

We remark that the estimates provided here were used in [14,15] to prove local existence for the system (3.3)–(3.6).

As the local existence proof is a little lengthier, we do not reproduce it here.

Lemma A.1.LetXbe a Banach algebra. Ifx∈Xis such thatxρ <1then1+xis invertible and(1+x)⁻¹

1

1−ρ. Further if in additionyρthen (1+x)⁻¹−(1+y)⁻¹ 1

(1−ρ)²x−y.

Proof. The first part of the lemma follows immediately from the identity(1+x)⁻¹=

(−x)ⁿ. The second part follows from the first part and the identity

(1+x)⁻¹−(1+y)⁻¹=(1+x)⁻¹(y−x)(1+y)⁻¹. 2 Lemma A.2.Ifk1, then there exists a constantc=c(k, α)such that

f ◦gk,αcfk,α

1+ ∇gk−1,α

k+α

and

f ◦g₁−f◦g₂k,αc∇fk,α

1+ ∇g₁k−1,α+ ∇g₂k−1,α

k+1

· g₁−g₂k,α.

The proof of Lemma A.2 is elementary and not presented here.

Lemma A.3.LetX₁, X₂∈C^k⁺^1,αbe such that

∇X₁−Ik,αd <1 and ∇X₂−Ik,αd <1.

LetA1andA2be the inverse ofX1andX2, respectively. Then there exists a constantc=c(k, α, d)such that A₁−A₂k,αcX₁−X₂k,α.

Proof. Letc=c(k, α, d)be a constant that changes from line to line (we use this convention implicitly throughout this paper). Note first∇A=(∇X)⁻¹◦A, and hence by Lemma A.1

∇AC⁰(∇X)⁻¹

C⁰c.

Now using Lemma A.1 to bound(∇X)⁻¹0,αwe have

∇A0,α=(∇X)⁻¹◦A

0,α(∇X)⁻¹

0,α

1+ ∇AC⁰

c.

Whenk1, we again bound(∇X)⁻¹k,αby Lemma A.1. Taking theC^k,αnorm of(∇X)⁻¹◦Awe have

∇Ak,α(∇X)⁻¹

k,α

1+ ∇Ak−1,α

k

.

So by induction we can bound∇Ak,αby a constantc=c(k, α, d). The lemma now follows by applying Lemma A.2 to the identity

A1−A2=(A1◦X2−I )◦A2=(A1◦X2−A1◦X1)◦A2. 2

Lemma A.4.Let u∈C([0, T], C^k⁺^1,α)and X satisfy the SDE (3.3)with initial data (3.6). Let λ=X−I and U=sup_tu(t )k+1,α. Then there existsT =T (k, α,^U_L)andc=c(k, α,^{U T}_L )such that fortT

∇λ(t )

k,αcU t

L and ∇ (t )

k,αcU t L hold almost surely.

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Proof. From Eq. (3.3) we have

X(x, t )=x+ t

0

u

X(x, s), s ds+√

2ν Wt⇒ ∇X(t )=I+ t

0

(∇u)◦X· ∇X. (A.1)

Taking theC⁰norm of Eq. (A.1) and using Gronwall’s Lemma we have ∇λ(t )

C⁰=∇X(t )−I

C⁰e^{U t /L}−1.

Now taking theC^k,αnorm in Eq. (A.1) we have ∇λ(t )

k,αc t

0

∇uk,α

1+ ∇λk−1,α

k+α

1+ ∇λk,α

.

The bound for∇λk,αnow follows from the previous two inequalities, induction and Gronwall’s Lemma. The bound for∇ k,αthen follows from Lemma A.3.

We draw attention to the fact that the above argument can only bound∇λ, and notλ. Fortunately, our results only rely on a bound of∇λ. 2

We conclude this appendix by stating a slightly modified version theorem which appeared in [14]. The only mod- ification we make is that we trace the dependence of the constants in [14] to dimension less quantities, instead of absolute ones. The proof that appeared in [14] goes through verbatim.

Theorem A.5. Let k1 and u0∈C^k⁺^1,α be divergence free. There exists absolute constants δ=δ(k, α, d)and C=C(k, α, d)such that forU=Cu0k+1,α, and anyT such that ^{U T}_L < δ there exist a pair of functions a pair of functionsλ, u∈C([0, T], C^k⁺^1,α)such thatuandX=I+λsatisfy the system(3.3)–(3.6). Further for allt∈ [0, T] we haveutk+1,αU.

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